Chapter 9.4, Problem 54SR

a.

To determine

To write the equation of the parabola formed by the mirror assuming the focus it at the origin.

Answer to Problem 54SR

  $y^2=39x+380.25$

Explanation of Solution

Given:

The focus of the mirror is $9.75$ feet above the vertex and the latus rectum is $39$ feet long.

Calculation:

The following is the equation of the parabola-

  ${(y-k)}^2=4p(x-h)$

Where, $\left(h+p,k\right)$ is focus and $4p$ is the length of latus rectum.

  $\begin{array}{l}\Rightarrow 4p=39\\ \Rightarrow p=\frac{39}{4}=9.75\end{array}$

It is given that

  $\begin{array}{l}\left(h+p,k\right)=(0,0)\\ (h+9.75,k)=(0,0)\\ \Rightarrow h=-9.75,k=0\end{array}$

So, the equation of parabola is-

  $\begin{array}{l}$ {(y-k)}^2=4p(x-h)$y^2=39(x+9.75)\\ y^2=39x+380.25\end{array}$

b.

To determine

To write the equation for the mirror in meters.

Answer to Problem 54SR

  $y^2=11.9x+116$

Explanation of Solution

Given:

One foot is $0.3048$ meter.

Calculation:

The equation of the parabolic mirror in meter is-

  $\begin{array}{l}{y}^2=(39x+380.25)\times 0.3048\\ y^2=11.9x+116\end{array}$

c.

To determine

To graph one of the equations of the mirror.

Explanation of Solution

Given:

The given equation of parabola is -

  $y^2=11.9x+116$

Calculation:

The following is the graph of the parabolic mirror-

  

d.

To determine

To explain which equation you choose to graph.

Explanation of Solution

Given:

The focus of the mirror is $9.75$ feet above the vertex and the latus rectum is $39$ feet long.

Calculation:

The equation that expressed in meter was chosen because it contains simpler terms than those expressed in feet.